Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Affine Deligne-Lusztig Varieties of Positive Coxeter Type

Felix Schremmer, Ryosuke Shimada, Qingchao Yu

Abstract

We introduce a class of affine Deligne--Lusztig varieties that we call of positive Coxeter type. We show that the affine Deligne--Lusztig varieties of positive Coxeter type have a very simple and explicitly described geometric structure. Conversely, we explain how some of these geometric properties can be used to characterize this class. These results vastly generalize the work of He--Nie--Yu on affine Deligne-Lusztig varieties of finite Coxeter type, leading to applications to Shimura varieties that were not possible using the old notion.

Affine Deligne-Lusztig Varieties of Positive Coxeter Type

Abstract

We introduce a class of affine Deligne--Lusztig varieties that we call of positive Coxeter type. We show that the affine Deligne--Lusztig varieties of positive Coxeter type have a very simple and explicitly described geometric structure. Conversely, we explain how some of these geometric properties can be used to characterize this class. These results vastly generalize the work of He--Nie--Yu on affine Deligne-Lusztig varieties of finite Coxeter type, leading to applications to Shimura varieties that were not possible using the old notion.
Paper Structure (22 sections, 31 theorems, 86 equations)

This paper contains 22 sections, 31 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Applications
  5. A Converse to Theorem \ref{['thm:introGeometry']}
  6. Preliminary
  7. Affine Weyl group
  8. $\sigma$-conjugacy classes
  9. Affine Deligne-Lusztig varieties
  10. Deligne-Lusztig reduction
  11. $P$-alcove elements and length positivity
  12. Generic Newton points and minimal Newton points
  13. Minimal Newton point of positive Coxeter type elements
  14. Quantum Bruhat graph
  15. The support of positive Coxeter type elements
  16. ...and 7 more sections

Key Result

Theorem A

Let $x=w\varepsilon^\mu \in\widetilde{W}$ and $v\in \mathop{\mathrm{LP}}\nolimits(x)$ such that $(x,v)$ is a positive Coxeter pair, that is, $\sigma^{-1}(v)^{-1} w v$ is a partial $\sigma$-Coxeter element. Let $J = \mathop{\mathrm{supp}}\nolimits_\sigma(\sigma^{-1}(v)^{-1} wv)$ denote its $\sigma$-s

Theorems & Definitions (65)

  • Theorem A: Cf. Theorem \ref{['thm:fctConsequences']}
  • Proposition B: Cf. Proposition \ref{['prop:pctCharacterization']}
  • Proposition 2.1
  • Definition 3.1: Schremmer2022_newton
  • Definition 3.2: Goertz2015
  • Definition 3.3: Schremmer2022_newton
  • Definition 3.4
  • Remark 3.5
  • Theorem 3.6
  • Lemma 3.7
  • ...and 55 more